The U.S. Centers for Disease Control and Prevention estimates that 300,000 sports-related concussions occur yearly in the U.S., but that number includes only athletes who lost consciousness. Since loss of consciousness is thought to occur in less than 10% of concussions, the actual number is probably closer to 3 million a year. Further, half of these concussions occur in children age 0 to 14, and an addition 38% occur in the age group 15 to 34. Only recently has the extent of sports-related mild traumatic brain injury (MTBI) become known among the general public. However, individuals and organizations concerned with protecting sports participants have pursued better protective headgear since the early 1940s. Such efforts have mostly focused on improving the cushioning inside the helmet's hard plastic shell. However, to date, no helmet has been able to completely eliminate concussions.
The laws of Physics limit the effectiveness of internal cushioning because of the limited space insides helmets. Current day football helmets weigh approximately four pounds and contain air cushioning that's approximately one inch thick. The helmets fit over athletes' heads, weighing an average of 11 pounds. A sharp blow that causes a 100 g acceleration (g being the acceleration of gravity, or 32.2 feet/sec2) of the helmet cannot be reduced to a safe level with just one inch of padding. Further, the force delivered to the head can be amplified during “rebound” conditions. Such conditions occur when the athlete's head is already moving in the direction of the force applied to the helmet, as what happens when the helmet bounces off the ground. This is what occurred in 1960 when Philadelphia Eagle Chuk Bednarik tackled NY Giant Frank Gifford, causing a life-long head injury that caused Frank Gifford to miss the entire next professional football season. Concussions resulting from head collisions are now being treated as a serious issue.
Most concussion studies have focused on measuring head accelerations rather than devising ways of reducing accelerations. Companies and organizations have developed small accelerometers that players may ware to record the accelerations (linear and rotational) experienced during contact. Such data is useful in determining the acceleration levels that produce concussions, currently believed to be between 100 g and 150 g.
The National Operating Committee on Standards for Athletic Equipment (NOCSAE) has meticulously developed methodologies to test the ability of football headgear to limit head accelerations. Their principal test for football helmets consists of dropping a helmeted Headform (i.e. simulated head) onto a half inch thick polyurethane pad that measures the deceleration of the helmeted Headform. The tests prescribe a variety of impact velocities with the helmet oriented in different positions. Further, the NOCSAE has developed a measure of the severity of impacts, called the Severity Index (SI). The SI is defined as the integral of the acceleration raised to the 2.5 power, measured over the interval when the acceleration rises above 4 g, and drops back below 4 g. Mathematically, this becomes:
  SI  =            ∫              t        1                    t        2              ⁢                  A        2.5            ⁢              ⅆ        t            
where A is the instantaneous acceleration, expressed as a multiple of the gravitational acceleration, g.
The NOCSAE prescribes limits on SI as a function of impact speed and point of impact on the helmet. As such, the Severity Index provides an objective measure of the protective value of helmets. For example, NOCSAE prescribes that the SI of a 17.94 foot/sec impact shall not exceed 1200 for any helmet impact orientation.
While prior designs that only employ cushioning material on the inside of the helmet can meet NOCSAE requirements, they cannot guarantee protection against concussions. Accordingly, investigators have looked at adding cushioning material on the outside of the helmet. In particular, Alfred Pettersen (US Patent 20150000013A1) invented an exterior sport helmet pad that was formed to fit over the helmet, and was held in place by internal contact pressure. Although such a cushioning device may provide some additional protection, it would be prohibitively expensive because a separate mold would be required for each size of each helmet design. Further, covering the entire outer surface of the helmet with extra padding could make the helmet excessively heavy, and thereby adversely affect athletic performance. In addition, Pettersen's conclusion that a pad thickness of only 0.5 to 0.75 inches thick was sufficient to mitigate concussions was unfounded because his tests did not conform to NOCSAE specifications. In particular, Pettersen's top impact speed was 8.97 feet/sec, which is below the prescribed NOCSAE minimum test speed of 11.34 feet/sec, and well below the NOCSAE maximum test speed of 17.94 feet/sec. Finally, Pettersen did not report the Severity Index for any test (as required by NOCSAE), but rather reported the amount that his external padding reduced the peak acceleration. Although Pettersen described the NOCSAE helmet tests within his patent, he gave no indication that any such tests were actually performed as specified.
An analytical model of the dynamics governing helmet impact can help methodically design a cushioning system capable of inhibiting concussions. Analytical models not only reveal the key parameters for optimizing helmet performance, but also enable the calculation of key performance metrics, such as the Severity Index, for example. Accordingly, an analytical model is presented within this document that reveals how accelerations and deformations are related to impact velocity and the geometry and properties of the cushioning materials. Further the model yields optimum values for padding thickness and stiffness, and provides quantitative estimates of the associated SI values. This system of equations can be used by future investigators to optimize helmet performance under various conditions.
A pad of cushioning material may be modeled as a spring, having an effective spring constant, Ke (pounds/foot). Ke is the force, F (pounds), divided by the amount, 6 (feet), that the pad is compressed. Mathematically:
      K    e    =      F    δ  
Also, according to Newton's Second Law, force equals mass times acceleration. Mathematically:F=Ma 
Combining these two equations to eliminate F yields an equation for acceleration in terms of displacement:
  a  =      δ    ⁡          (                        K          e                M            )      
The spring constant, Ke, is a function of the pad's elastic modulus, E (pounds/square inch), contact area, “Area” (square inches), and thickness, L (feet). Hence:
      K    e    =                    (        Area        )            ⁢      E        L  
Combining the above two equations yields the required elastic modulus, Ereq, to arrest a particular acceleration:
      E    req    =            L      δ        ⁢          M              (        Area        )              ⁢    a  
For example, for a mass, M, consisting of a head weighing 11 pounds, and a helmet weighing 4 pounds, the Elastic Modulus required to arrest a 100 g acceleration, using a one-inch thick pad, 4 inches in diameter, and compressed to half its thickness is given by:
      E    req    =                              (                      1            /            12                    )                          (                      1            /            24                    )                    ⁢                        (                      11            +            4                    )                          (                      4            ⁢            π                    )                    ⁢      100        =          238.7      ⁢                          ⁢      psi      
The corresponding maximum force exerted on the helmet would equal Mass times acceleration or 1500 pounds (100 g×15#/g).
We can determine the impact velocity, VImp, required to produce a 100 g acceleration by using the conservation of energy principal. For an inelastic collision, with a coefficient of restitution, ε, between the colliding bodies, we can equate the Potential Energy to the Kinetic Energy, less the energy lost in the collision. The Coefficient of Restoration, ε, accounts for the energy dissipated in the collision. Hence we have:
                    K        e            ⁢              δ        2              2    =                    (                              M            s                    +                      M            h                    +                      M            p                          )            ⁢                        (                      ɛ            ⁢                                                  ⁢                          V              Imp                                )                2              2  
where Ms, Mh, and Mp are the masses of the shell, head, and pad, respectively. Collectively, these masses equal the total mass, Mt. Solving for VImp yields
      V    Imp    =            δ      ɛ        ⁢                            K          e                          M          t                    
However, the maximum linear acceleration is not the sole determinant of injury level—the variation of that acceleration with time also influences the severity of an impact. This time variation for a simple spring-mass system is given by:
      a    =                            V          Imp                ⁢        ɛ        ⁢                                            K              e                                      M              t                                      ⁢                  sin          ⁡                      (                                                                                K                    e                                                        M                    t                                                              ⁢              t                        )                              =                        V          Imp                ⁢        ɛω        ⁢                                  ⁢                  sin          ⁡                      (                          ω              ⁢                                                          ⁢              t                        )                                ,
where
  ω  ≡                    K        e                    M        t            
We can now substitute this equation for acceleration into the SI equation to estimate the severity of collisions. Since the SI equation has an acceleration exponent of 2.5, a precise calculation of SI requires a numerical integration. However, a reasonable analytical approximation for SI may be obtained by integrating A2 instead of A2.5, and raising the result of the integration to a power of 1.25 (2×1.25=2.5). This produces a number that is typically 5% greater than the integral of A2.5. Hence the results should be multiplied by a factor of 0.95. Thus an approximate value of SI is given by:
  SI  ≅      0.95    ⁢                  (                              ∫                          t              ⁢                                                          ⁢              1                                      t              ⁢                                                          ⁢              2                                ⁢                                    A              2                        ⁢                          ⅆ              t                                      )            1.25      
where t1 is the time when the acceleration first reaches 4 g, and t2 is the time when the acceleration decays back down to 4 g. For our sine function, t2=π/ω−t1. Carrying out the integration yields the following equation for estimating SISI≅0.95[0.5ω(εVImp/g)2(sin(2β)+π−2β)]1.25 
where
  β  ≡      ARC    ⁢                  ⁢          SIN      ⁡              (                              4            ⁢                                                  ⁢            g                                ωɛ            ⁢                                                  ⁢                          V              Imp                                      )            
Substituting the numbers used in the example calculation of E yields the following tabulation of Severity Index versus impact velocity:
VImp (ft/sec)11.3413.8916.0417.94ω = SQRT(Ke/(Ms + Mh)) (radians)279.02279.02279.02279.02β = ARCSIN(4 g/(ωεVImp)) (radians)0.0820.0670.0580.051Calculated Severity Index (SI)24.7841.1558.9878.02NOCSAE SI Limits300.001200.001200.001200.00
The calculated SI values are at least an order of magnitude below the NOCSAE limits. Further, when tested, the actual SI values will be reduced from the calculated values because of the cushioning inside the helmet. Accordingly, we can relax the example requirement on E by approximately a factor of two to 120 psi, and still have a quite viable concussion mitigation configuration. An E value of 120 psi is well within the softness of commonly available soft rubbers, such as neoprene and silicone. Polymer foam and sponge materials are also commonly available that have elastic moduli in this range. For example, Saint-Gobain's CORHlastic® 300/9030 Silicone solid rubber has an E value of 225 psi, and the company's firm sponge rubber has an E value of 111 psi.
It should be pointed out that there is some evidence that rotational accelerations of the head also play a role in concussions. However, the external cushioning system that mitigates linear accelerations should also mitigate rotational accelerations.
Since we have established that soft rubber pads affixed to the outside of helmets are capable of mitigating injuries, the issue now becomes how to best attach such cushioning material to headgear. There are three ways of attaching cushions to helmets: gluing (or bonding); form-fitting; or mechanically anchoring. Gluing or bonding pads to the outside of helmets is problematic for several reasons. First, it is difficult to get a planar-shaped pad to conform to a three-dimensional ellipsoidal-shaped helmet. Forcing such conformance creates high residual stresses in both the bonding interface and the pad material itself that will break over time. Second, sharp blows can dislodge glued pads from the helmet. Third, gluing is a time-consuming and craft-sensitive operation which would add considerable cost to externally padded helmets.
Form-fitting also has three major disadvantages. First, it would be prohibitively expensive to build custom molds for all existing and future helmet designs, and for all head sizes, because the lowest quotes for molds to produce such products are well over a thousand dollars. Also, the material cost of a full helmet cap is roughly three times the cost of a single four-inch wide strip covering just the crown of the helmet. Second, covering the entire outer surface of the helmet with material thick enough to appreciably reduce impact forces would add excessive weight that could adversely affect athletic activities. Finally, a fully conforming helmet cover would create an environment between the pad and the helmet that is conducive for the growth of mold, bacteria, and other pathogens that could be harmful to the user.
For the above reasons, the best choice is to mechanically anchor padding material to headgear. However, such anchoring must be done without employing hard protruding objects, such as brackets and buckles, which can pose injury risks. Also, the mechanical mechanism must keep the pads securely attached to the helmet through environmental extremes (temperature and humidity), and during severe blows. Further, the mechanism must be lightweight, inexpensive, and easily installed. Finally, the mechanism must work on all helmets, and force all pads to conform to the shape of the helmet.
A universal solution is to use a strap that runs over the outermost surface of the pad, and that anchors to existing features (or elements) of the helmet. Examples of such features include the facemask lattice (or other protrusions), holes through the helmet, and edges of the helmet. The securing straps can come in a variety of shapes and materials, such as netting that covers the entire outer surface of the pad, or ropes, or thin, flat, narrow belts that run down the centers of the pads. Such straps can be either separate components, or material permanently bonded to the pads. The flexible pads may be either planar-shaped planks (or pads), or curved planks whose shape roughly matches that of most helmets. The flexible pads may also be air-filled tubes or chambers (either connected or individual members), or structures composed of different materials. Finally, the anchoring devices may range from the strap itself, as in the case of self-anchoring Velcro® tapes, to special hooks that resemble present day fishing hooks (with a loop for the fishing line, but without the barb on the end). Anchoring devices may also range from small commercially-available clips and buckles, to small loops or brackets that are riveted or otherwise affixed to the helmet. Hard anchoring devices are acceptable provided they don't protrude appreciably from the shell, or are covered by soft material. Small buckles affixed to straps running down the center of a pad will be pushed down into compliant cushioning material (along with the rest of the strap), by the high tape tensile forces that indent the cushioning material.